Heat your grill to medium/medium-high. Cut the radicchio through the root end so each quarter has one fourth of the core. This will help keep each segment in tact. If any of the outer leaves do detach, simply grill them separately. Rinse the radicchio quarters, pat dry and set aside.

Scrape the membrane off the squid bodies. Cut down one side of each body and unfold. With a paring knife, make a series of cuts, less than halfway into the flesh, along the length of the body. Now make another series of cuts diagonally crossing the first ones. Again, be very careful to not cut too deep or the squid will come apart. This procedure is known as scoring and will allow them to grill flat and not curl up.

Brush the squid and the radicchio quarters with olive oil. Start by grilling the radicchio on one of the flat sides. Each side will take 2-3 minutes. Give them a head start and then start grilling the squid, turning once, until just cooked through and lightly golden, about 2-3 minutes. Do not overcook or it will become rubbery.

Mix the arugula and red onion and place in the center of a large platter. Cut the calamari into strips and sprinkle over top of the arugula. Arrange the radicchio quarters around the pile of arugula. Whisk the olive oil, lemon juice, garlic and salt and pepper and drizzle over the salad.